An ellipse has OB as a semi-minor axis, F, F' as its foci and the angle ∠FBF′ is a right angle. Then the eccentricity of the ellipse is

A
$\frac{2}{3}$
B
$\frac{1}{\sqrt{2}}$
C
$\frac{3}{5}$
D
none

Solution:

We have

$\begin{array}{l} ∠ F B F^{'} = \frac{π}{2} \\ ∴ F F^{' 2} = F B^{2} + F^{'} B^{2} \\ \Rightarrow (2 a e)^{2} = a^{2} + a^{2} [\begin{matrix} ∵ x -coordinate of B is zero \\ ∴ F B = a - e \times 0 and F^{'} B = a + e \times 0 \end{matrix}] \\ \Rightarrow 4 e^{2} = 2 \Rightarrow e = \frac{1}{\sqrt{2}} \end{array}$

An ellipse has OB as a semi-minor axis, $F, F$ ' as its foci and the angle $∠ F B F^{'}$ is a right angle. Then the eccentricity of the ellipse is

Solution:

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